Physics 172 – Recitation 06

Spring 2010

As you work on today’s problems remember that every time you use a fundamental principle you must explain clearly what physical system you are applying the principle to and which objects in the system’s surroundings are interacting significantly with it.

Problem 1. This figure shows the positions from 1995 to 2004 of a star, S0-20, that orbits the center of our galaxy. The orbit is nearly circular with the radius shown.

For more detail on how the remarkable observations of this star’s position were made and on black holes at the centers of galaxies see 5.P.66 on page 217 of the text.

a) Using the positions and times shown, estimate the stars speed. Express your answer in MKS units and as a fraction of the speed of light.

[From the figure, it looks like the star moved about a quarter of the way around its circular path between 1995 and 2003, that is, it moved about

in about . So, the star’s speed is about

. ]

b) This is an extraordinarily high speed for a macroscopic object. Is it, nevertheless, reasonable to estimate the star’s momentum as ?

[It is because for the star’s speed

. ]

c) Construct a simplified model of the star’s surroundings and use it to estimate the mass of the black hole about which S0-20 is orbiting.

[Let’s simply ignore the presence of everything in the star’s neighborhood but the black hole. In this case, the gravitational force exerted on the star by the black hole located at the center of the star’s circular orbit must account for the rate at which the star’s momentum is changing. The perpendicular component of the momentum principle applied to the star (the “centripetal component” toward the center of the star’s orbit) is

where M is the mass of the black hole and m is the mass of the star. This implies that

]

d) How does your estimate of the black hole’s mass compare to the Sun’s mass, ? It is thought that all galaxies may have such a large black hole at their centers, the result of mass accumulation at a galaxy’s center since the galaxy’s formation.

[Our result from part c implies

A black hole, a compact type of object predicted by Einstein’s general relativity, is the only kind of object that could account for such an enormous amount of matter in such a small region (speaking astronomically) of space. ]

[Checkpoint 1]

As you solve the next problem, notice that the Energy Principle and the definitions of work and particle energy make it possible to solve problems in new and interesting ways.

Sometimes applying either the Momentum or Energy principle to a system will build equations that you can solve to answer questions about the system in its surroundings. As you will see in this problem, applying one or the other will lead more directly to the answer.

As you gain experience solving problems, you will get better and better at planning simple solutions by choosing to use physics principles in the most efficient ways.

Problem 2.

a) This diagram depicts the initial positions of two disks on a frictionless surface. Disk 2 is four times as massive as disk 1. Starting from rest, the disks are pushed across the surface by two equal forces.

Which disk has the greater kinetic energy after one second? Explain your reasoning.

b) This diagram depicts the initial positions of two disks on a frictionless surface. Disk 2 is four times as massive as disk 1. Starting from rest, the disks are pushed across the surface by two equal forces.

Which disk has the greater kinetic energy when it crosses the finish line? Explain your reasoning.

[Answer a)

Because equal forces act on each object for 1 second, it is useful to apply the momentum principle to each disk. Doing so and using the fact that both disks were initially at rest, we conclude that their momenta are equal after one second, . Treating the disks motions non-relativistically, we have

where and are the disk’s speeds after one second. Evaluating the disks kinetic energies we find that

Alternatively, because this is a qualitative question, we could simply have noted that the equal forces on the two disks will cause the less massive disk, disk 1, to move further in one second than the more massive disk does. This means that the force on that less massive disk does more work on it than the force exerted on the more massive disk does. Applying the energy principle to each disk then leads us directly to the conclusion that, since no other forces do work on either disk system and no significant flow of thermal energy to or from either disk system occurs.

Answer b)

In this case, it natural and simplest to apply the energy principle to each of the disks we know that the equal forces exerted on each disk act through the same distance. This means that the forces do equal work on each of the disks. Since both started from rest, the kinetic energy that each has as it crosses the finish line will be equal.

Notice, however, that the disks do not cross the finish line at the same time! The less massive disk will reach the finish line first. ]

[Checkpoint 2]

Problem 3. A mass of 0.12 kg hangs, at rest, from a vertical spring in the lab room. At time t = 0 you hit the mass so that begins to move straight downward. Its speed of the mass just after you strike it is 3.40 m/s.

a) While the mass moves downward a distance of 0.07 m, how much work is done on the mass by the Earth?

[While the mass moves straight down 0.07 m the Earth is exerting a gravitational force straight down on the mass with an essentially constant magnitude of

Since the force and displacement are parallel, the work done by this force is positive and it amounts to . ]

b) If the speed of the mass is 2.85 m/s at that point, estimate how much work was done on the mass by the spring?

[If we apply the energy principle to the system consisting of the mass, we will obtain an equation that involves the thing we are asked about – the amount of work done on the mass by the spring as the mass moves downward 0.07 m. The equation will also involve the change in the mass’s kinetic energy, which we can determine because we know the mass’s initial and final speeds, and the amount of thermal energy transferred to the mass, which we can assume is negligible. Consequently, applying the energy principle to the mass should allow us to solve this problem.

Applying the energy principle to the mass and neglecting thermal effects yields the equation

Plugging in what we know and making the reasonable non-relativistic approximation, we find that

]

[Checkpoint 3]